Find the missing side of a right triangle.
This Pythagorean theorem calculator finds the missing side of any right triangle. Choose whether you need the hypotenuse or one of the legs, enter the two sides you know, and you get the answer with the working, plus both acute angles, the area and the perimeter.
Quick answer: in a right triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). To find the hypotenuse, c = √(a² + b²). To find a leg, a = √(c² − b²). A triangle with legs 3 and 4 has a hypotenuse of 5.
How to Use the Pythagorean Theorem Calculator
- Under Find, choose Hypotenuse c, Leg a or Leg b.
- Enter the two sides you know. Use the same unit for both.
- Read the missing side, angle A and angle B in degrees, the area and the perimeter.
The answer is in the same unit you used for the inputs.
The Pythagorean Theorem Formula
a² + b² = c²
- a and b are the two shorter sides (the legs), which meet at the right angle.
- c is the hypotenuse, the longest side.
Rearranged:
- Hypotenuse: c = √(a² + b²)
- Missing leg: a = √(c² − b²)
Pythagorean Theorem Examples
Find the hypotenuse. Legs 3 and 4: c = √(9 + 16) = √25 = 5. The angles are about 36.87° and 53.13°.
Find a leg. Hypotenuse 13, one leg 5: a = √(169 − 25) = √144 = 12.
A ladder against a wall. A 10 ft ladder stands with its base 6 ft from the wall. It reaches √(100 − 36) = √64 = 8 ft up the wall.
A room’s diagonal. A 9 ft × 12 ft room has a diagonal of √(81 + 144) = √225 = 15 ft.
A square’s diagonal. A square with sides of 1 has a diagonal of √2 ≈ 1.414.
Common Pythagorean Triples
These whole-number sets fit a² + b² = c² exactly. Any multiple of a triple also works, such as 6, 8, 10.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 12 | 15 |
| 10 | 24 | 26 |
Builders use the 3-4-5 triple to check that a corner is square: measure 3 units along one side and 4 along the other, and the diagonal should be exactly 5.
Mistakes to Avoid
- Using it on a triangle without a right angle. The theorem only works for right triangles.
- Putting the longest side in the wrong place. The hypotenuse is always c, opposite the right angle.
- Adding instead of subtracting for a leg. For a missing leg, subtract: c² − b².
- Forgetting the square root at the end. √25 = 5, not 25.
Related Math Tools
- For square roots and simplified radicals, such as √50 = 5√2, use the square root calculator.
- To check a calculation such as √(3^2+4^2) yourself, use the scientific calculator.
- For more geometry formulas, see our list of basic math formulas.
Pythagorean Theorem FAQs
Does the Pythagorean theorem work for all triangles?
No, only for right triangles. For other triangles, use the law of cosines, which a triangle calculator handles.
How do I know which side is the hypotenuse?
It is the longest side, and it is opposite the right angle. It never touches the right angle.
How do I check whether a triangle is a right triangle?
Square the two shorter sides and add them. If the total equals the longest side squared, the triangle has a right angle. 5, 12, 13 works because 25 + 144 = 169.
Can I use different units?
Convert both sides to the same unit first, such as all centimetres or all feet. The answer will be in that unit.
How we checked this page: every example and triple was checked by hand and matches the calculator, including the angles.